A second question of interest concerns the “efficiency gain" of the multiple approach compared to the univariate one. In this sense, we would like to test whether taking into account the association structure among the market indices would provide us with better predictions in terms of VaR and ES. To do that, we use the backtesting procedure to identify the most “efficient” model, that is, the model producing the best forecasts according to the [1]} test. To measure the efficiency gain we analyze the difference, if any, in the predictive accuracy between the forecast (VaR, ES) produced by our multivariate approach and the univariate ones.
Therefore, for a given CAViaR specification, we test for the difference between the scores obtained with the scoring rule in (REF ) and the ones obtained with (REF ). The null hypothesis is that, on average, the difference is not statistically different from zero, i.e. the two approaches have the same forecasting performance. The alternative hypothesis is that the difference is smaller than zero, i.e. the multivariate approach delivers significantly better predictions (smaller losses).
